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arXiv:cond-mat/0411758 · ENLIGHTEN (Jurnal Bimbingan dan Konseling Islam)

Representation Theory and Baxter's TQ equation for the six-vertex model. A pedagogical overview

Christian Korff

Published Nov 30, 2004• 3 citations
arXivHTMLPDF

Abstract

A summary of the construction procedure of generalized versions of Baxter's Q-operator is given. Illustrated by several figures and diagrams the use of representation theory is explained step-by-step avoiding technical details. The relation with the infinite-dimensional non-abelian symmetries of the six-vertex model at roots of unity is discussed and parallels with the eight-vertex case outlined.

DOI
10.48550/arxiv.cond-mat/0411758

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A summary of the construction procedure of generalized versions of Baxter's Q-operator is given. Illustrated by several figures and diagrams the use of representation theory is explained step-by-step avoiding technical details. The relation with the infinite-dimensional non-abelian symmetries of the six-vertex model at roots of unity is discussed and parallels with the eight-vertex case outlined.
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We provide two methods of producing the QQQ-operator of XXZ spin chain of higher spin, one for NNNth root-of-unity qqq with odd NNN and another for a general qqq, as the generalization of those known in the six-vertex model. In the root-of-unity case, we discuss the functional relations involving the constructed QQQ-operator for the symmetry study of the theory. The QQQ-operator of XXZ chain of higher spin for a generic qqq is constructed by extending Baxter's argument in spin-1/2 case for the six-vertex QQQ-operator.
We construct the explicit Q -operator incorporated with the sl 2 -loop-algebra symmetry of the six-vertex model at roots of unity. The functional relations involving the Q -operator, the six-vertex transfer matrix and fusion matrices are derived from the Bethe equation, parallel to the Onsager-algebra-symmetry discussion in the superintegrable N -state chiral Potts model. We show that the whole set of functional equations is valid for the Q -operator. Direct calculations in certain cases are also given here for clearer illustration about the nature of the Q -operator in the symmetry study of root-of-unity six-vertex model from the functional-relation aspect.